(Edited for clarity.)
Take the primorial of the $n$th prime $p_n$ by using $H=\prod_{i=1}^np_i$.
Does there exist an $n$ such that there exists $d\mid H$ where there are no factors of $d^{p_{n+1}-1}$ which are primitive roots modulo $p_{n+1}$ and also no factors of $\left(\frac{H}d\right)^{p_{n+1}-1}$ which are primitive roots modulo $p_{n+1}$?
If such an $n$ were to exist, then some ideas regarding representation of congruence classes modulo $H$ would become more difficult to approach, such as the idea that a $Q:(Q,H)=1$ has a representation as a binary sum of products of factors of $H$, in particular for $d\mid H$ there might exist $r_1,r_2,\dots,s_1,s_2,\dots$ where $Q\equiv \prod_{u\mid d}u^{r_u}+\prod_{v\mid \frac Hd}v^{s_v}\mod H$.